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Plastic deformation of polycrystalline materials is largely controlled by the interaction between lattice dislocations and grain boundaries. The atomistic details of these interactions are, however, difficult to discern even by advanced high-resolution electron microscopy methods. In this paper, we study several interactions of screw and edge dislocations with two symmetric tilt grain boundaries in the body-centred cubic metal tungsten by atomistic simulations. Two distinct models of interatomic interactions are applied: an empirical Finnis-Sinclair potential and a bond-order potential, which is based on quantum mechanical principles within the tight-binding electronic-structure theory. Our study shows that the outcome of the interactions is sensitive to the employed interatomic potential. The origins of the deviating behaviour can be traced to differences in the description of atomic bonding by the two potentials. Independent of the employed interatomic potential, the simulations reveal that simple empirical criteria for dislocation transmission, which are based on geometry and stress arguments only, do not apply in general. Instead, in most cases, processes occurring at the atomic level play a decisive role in the determination of the underlying mechanisms of dislocations/grain-boundary interactions.